The direction cosines of the vector $\hat{i} + \hat{j} + \sqrt{2}\hat{k}$ are:

State with reasons,whether the following algebraic operations with scalar and vector physical quantities are meaningful:
$(a)$ adding any two scalars,
$(b)$ adding a scalar to a vector of the same dimensions,
$(c)$ multiplying any vector by any scalar,
$(d)$ multiplying any two scalars,
$(e)$ adding any two vectors,
$(f)$ adding a component of a vector to the same vector.

The vectors $(A + B)$ and $(A - B)$ are at right angles to each other. This is possible under the condition

Calculate the shortest wavelength in the hydrogen spectrum emission of the Lyman series $(R_H = 109677 \text{ cm}^{-1})$.

Find $|\Delta \overrightarrow{a}|$ and $\Delta a$ when a vector $\overrightarrow{a}$ is rotated by an angle $d\theta$.

Two vectors $A$ and $B$ are inclined at an angle $\theta$. Their resultant $R$ makes an angle $\phi$ with vector $A$. If the directions of $A$ and $B$ are interchanged,the resultant will have the same: